#### Read Microsoft Word - Phys560Notes-2.doc text version

`Sommerfeld Theory of Metals ­ a quantum theory of independent free electrons Physical picture: Focus on one electron. Assume all other charges in the system are smeared out into a static neutral background (atomic details, fluctuations, &amp; correlations ignored). All electrons are &quot;equal.&quot; A simplified jellium model. Main difference from Drude model ­ Fermi Dirac statistics. Consider an electron gas in a cubic box of volume V = L3 . Two commonly used boundary conditions: 1. Infinite well,  = 0 outside V; standing wave solutions; convenient for some problems. 2. Periodic boundary condition (PBC), or Born-von Karman BC:  x  L, y , z     x, y  L, z     x , y , z  L     x , y , z 1D example:Wave function repeats.We use PBC ­ convenient for mode counting. It is unphysical in 2D and 3D, but bulk properties are independent of V and boundary conditions as V   .1 Plane wave solution:  k  r   V exp  ik  r k = wave vector; quantum numberp k  r   i k  r   k k  r H k  r  It is a momentum eigenfunction.2 k 2 p2  k r    k  r     k  k  r  2m 2m k  2 k 2 2m(dispersion relation)PBC: exp ik x  x  L   ik y y  ik z   exp  ik x x  ik y y  ik z z   exp  ik x L   exp  ik y L   exp  ik z L   1k x, y ,z  2 nx , y , z Letc.nx , y , z   integersA dense grid of points in k space.3  2  8  Volume each k point  L  V   3V Density of points in k space = 8 3kV dk 8 3 for V  1 exp  ik  r     k r     Adding spin, V    k ,s2V dk 8 3 Ground state at T = 0 (minimum energy) based on Pauli exclusion principle:  2     2        g  A    0,0,0      0,0,0      ,0,0      ,0,0       L     L    2     2     2     2      L ,0,0      L ,0,0      0, L ,0      0, L ,0    ...              A = Antisymm operator g = a Slater determinantTotal energy is a minimum occupied k values are enclosed in a sphere in k space (Fermi sphere; Fermi surface; radius = Fermi wave vector)Nk k F 2  8 32Vdk k k F2V 4 3 V 3  kF  2 kF 3 8 3 3nN k3  F2 V 3k F   3 2 n 13~ 1 Å-1 for typical metallic density.F TF F2 2kF (Fermi energy) ~ 1.5 ­ 15 eV, typically. 2mkB(Fermi temperature) ~104 ­ 105 K, typically (&gt;&gt; RT).vF  2 F m (Fermi velocity) ~108 cm/s, typically (relativistic effects small).Ground state properties at T = 02V Total energy E  k 2  k   8 3 kFk  kF2V 3 2k 2 3 2k 2 d k 3  4 k 2 dk  N  F 2m 8 k  kF 2m 5 E 3 (average energy per particle)   F N 5From first law of thermodynamics, TdS  dU  pdV   dN  0 2 2  E   3  E     3  p   NF   N   3 2 n  3   ...  2 V  2 n F (not measurable)    3 5  V  N V  5  V  5 2 m p 2 Bulk modulus B  V V  3 n F ~1010 ­ 1012 dyne/cm2, typically; about the right order of magnitude for metals.Properties at T &gt; 0 Fermi Dirac distribution (probability of occupancy)0  f     1 (exclusion principle) f  , T       T   exp   1  kT  1   T  chemical potential = Gibbs free energy per particle; determined by normalization  f   k    Nk,s1 for     0  f  , T  0   f0   0 for     0 f0 0   F1For T &gt; 0, f ( , T )   1  2FkT  F ~ 0.01 at RT for typical metals.f(T) 1 1/2~kTOK to assume kT   F (low temperature limit)f ( , T ) ~ broadened negative step function, width ~kTd f ( , T )        ~ broadened delta function, width ~kT d  kT 2   T    F  O    F  ~  F for free electrons in 3D, to be verified (different for 2D &amp; 1D).   F    Density of states (levels) (per unit volume) g    1 g    d    # of one-electron levels between  and   d   V 1 Very useful for calculating  F    k     F    g    d  , where F is a function of energy. V k ,s 1 Summation in k space converted into an integral over energy.    d  g   V k ,s 1 2V 1 1 dk d 3 k  3  4 k 2 dk  2  k 2 d 3  V 8 4 d  k ,s 2 k 2 d 2k For free electrons,   .  dk m 2m 1 V1 Vk ,s12m k d    d  g   2 2 g   mk 2 m3   2 3    2 2  g  F  mk F density of states at the Fermi level, relevant to many properties.  2 2g   f  ,T T=0g   f  ,T T&gt;0Fn F N   f   , T g    d    g    d  = area under the curve, independent of T. 0 0 V This constraint yields  T  . Pictorially, it is clear that  must be slightly less than F in order to conserve the area.Define g     0 for   0 . Defineg   d   G   .n   d  f   , T  g    d   G    f   , T     G      f  , T  d   d    1 2  d  Expansion: n   G          g          g      ...   f  , T  d   2    d 1st term = G      f     d   G     f     f      G       2nd term = g     3rd term       f      d   0 , because the integrand is odd about .  5th term  O T 4  ; ...2 26 6 kT 2g     ; 4th term = 0 (odd integrand);2n  G    kT g      O T 4  G  F   G    (1)n T  0   n T  G     G  F   26 kT 2g      O T 4 26 kT  2 g      O T 4 1 22Expand:    F  g  F       F 2It is clear that    F  O T6 if only the first term is kept. The second term  O T 4  .g   F     2 kT 2g      O T 4   F 26 kT 2g   F  g  F  O T 4    F  O T 2 Whew! (Sommerfeld expansion)Specific heat (at constant volume)TdS  dQ  dU  pdVc1  dQ  1  U   u         V  dT V V  T V  T VRepeat the above derivation with g   g , see Eq. (1) previous page.u c g   f  , T  d     g   d  026 kT 2 g      g       O T 4   u d  2 2 2 d 2 3  g   k T  g      g        6  kT   2 g       g      dT  O T    T dT 3   2 2 g   F   2 2  O T 3    c   g     k T k T  g      g       O T 3    g  F   3  3 2 g   F    2 2 c   F g   F    k 2T k T  g   F    F g    F    O T 3     g  F   3  3c23k 2T g   F   O T 3  mk F ,  2 2With g   F  c 2  kT    nk 2  F Cf. cDrude 3 nk 2cS kT   0.01 at R.T. cD  FFor metals at RT, c is dominated by phonon contributions: c phonon  3 nk (verified later). Below a few K, c is dominated by electronic contributions (phonons are quenched out). Results in fairly good agreement with experiment (within 2x, typically) Why is cS  cD ?f(T) 1 1/2 kT  Number of electrons excited  n    F  Energy for each excited electron  kT~kT kT  unF2 kT So, c    F  nk For the Drude model, each electron has thermal energy  kT . So,cS kT   0.01 cD  FConsequences of different statistics: Physical Properties Volume specific heat c DC conductivity  o AC conductivity    Dielectric function    Hall coefficient RH Magnetoresistance   B  Thermal power Q v2 Drude 3 nk 2ne2 m  o 1  i  1  4 i  1 nec noneSommerfeld  2  kT    nk 2  F  Same* Same* Same* Same* Same* Same* 2 2  vF  F m ~108 cm/s ~100 Å Same*Ratio (D/S) ~100 at RT 1 1 1 1 ~100 at RT ~0.01 at RT ~0.1 at RT ~0.1 at RT ~1 (cancellation)c / 3ne 3kT / m~107 cm/s v   v (with  deduced from  ~10 Å Thermal conductivity K v2  c 3*Formulas are the same as those for the Drude results because we can still use the same d p ave p basic equation   ave  f  dt  = phenomenological relaxation time = relaxation time for distribution function f to relax back to f oFermi sphereFermi sphere displaced by an electric fieldDisplacement in the steady state with a field: p0   f ave Removing the field, = time to return to initial distributionp ave  p0 exp  t /   aveThis picture illustrates that n in   kT  ne 2 is the total electron density, not  n  . m  F `

6 pages

#### Report File (DMCA)

Our content is added by our users. We aim to remove reported files within 1 working day. Please use this link to notify us:

Report this file as copyright or inappropriate

625140

Notice: fwrite(): send of 197 bytes failed with errno=104 Connection reset by peer in /home/readbag.com/web/sphinxapi.php on line 531